Evaluate the following
`cos [ cos^(-1) ( (-1)/3) - sin^(-1) ( 1/3)]`.

Answer

To evaluate the expression \( \cos \left[ \cos^{-1} \left( -\frac{1}{3} \right) - \sin^{-1} \left( \frac{1}{3} \right) \right] \), we can follow these steps:

### Step 1: Rewrite the expression using properties of inverse functions
We know that:
\[ \cos^{-1}(-\theta) = \pi - \cos^{-1}(\theta) \]
Thus, we can write:
\[ \cos^{-1} \left( -\frac{1}{3} \right) = \pi - \cos^{-1} \left( \frac{1}{3} \right) \]

### Step 2: Substitute this into the original expression
Now substituting this into our expression, we have:
\[ \cos \left[ \pi - \cos^{-1} \left( \frac{1}{3} \right) - \sin^{-1} \left( \frac{1}{3} \right) \right] \]

### Step 3: Use the identity for cosine of a difference
Using the cosine of a difference identity:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
Let \( A = \pi \) and \( B = \cos^{-1} \left( \frac{1}{3} \right) + \sin^{-1} \left( \frac{1}{3} \right) \). Thus, we have:
\[ \cos \left[ \pi - (\cos^{-1} \left( \frac{1}{3} \right) + \sin^{-1} \left( \frac{1}{3} \right)) \right] = -\cos \left( \cos^{-1} \left( \frac{1}{3} \right) + \sin^{-1} \left( \frac{1}{3} \right) \right) \]

### Step 4: Simplify using the identity for inverse functions
We know that:
\[ \cos^{-1} \theta + \sin^{-1} \theta = \frac{\pi}{2} \]
Thus:
\[ \cos^{-1} \left( \frac{1}{3} \right) + \sin^{-1} \left( \frac{1}{3} \right) = \frac{\pi}{2} \]

### Step 5: Substitute back into the expression
Now we can substitute this back:
\[ -\cos \left( \frac{\pi}{2} \right) \]

### Step 6: Evaluate the cosine
We know that:
\[ \cos \left( \frac{\pi}{2} \right) = 0 \]
Thus:
\[ -\cos \left( \frac{\pi}{2} \right) = -0 = 0 \]

### Final Answer
Therefore, the value of \( \cos \left[ \cos^{-1} \left( -\frac{1}{3} \right) - \sin^{-1} \left( \frac{1}{3} \right) \right] \) is:
\[ \boxed{0} \]